This section discusses the factors influencing the position of a train on a curved railway, crucial for determining the appropriate track radius in developing the curved track G2T-FSO link model. Since the advent of railways in 1825, the basic systems for straight and curved track navigation have remained consistent. Unlike other vehicles, trains are guided by their wheelsets along the track without active steering by a driver [14]. On curved tracks, centrifugal forces cause the train to tilt. Key parameters include the cant angle (δ-Cant), track cant (U), and track gauge (G), as shown in Fig. 1. Residual lateral acceleration (ζ) is measured in m/s² or g, while cant excess (E) and cant deficiency (I) are measured in millimeters.

In railway engineering, proper distribution of train load on wheelsets is essential, particularly on curved tracks with a specific cant (or superelevation). This design helps reduce wear and tear between the rails and rolling stock, counteracting the effects of lateral forces [15]. These measures ensure passenger comfort and prevent derailment by minimizing the risk of wheel slippage [16]. Additionally, trains must maintain an equilibrium speed when negotiating curves to balance train load forces and radial acceleration, keeping them perpendicular to the track plane. Cant helps counteract centrifugal forces at this equilibrium speed, ensuring stability and smooth operation [16].

Cant deficiency occurs when the train’s speed exceeds the equilibrium speed, causing it to tilt towards the outer rail of the curve. Conversely, if the speed is lower than the equilibrium speed, cant deficiency causes the train to tilt towards the inner rail. To calculate the maximum permissible speed for a curve in this paper, parameters such as track cant length, cant excess, cant deficiency, and track radius must be considered. Fig. 2 illustrates the track curvature for a single wheelset, with V representing the train’s speed along the curve [17].

Fig. 3 illustrates a curved track with G2T-FSO communication, where the link connects to ground base stations (BSs) along the railway. Multiple FSO transceivers on the train’s roof enable line of sight (LOS) communication with the BSs. ZC1 is the propagation distance between the FSO transceiver and the BS, while θC1 and θC112 represent the divergence and half-divergence angles, respectively.

The focus of this paper is on high-speed trains. They have no clear definition but are often recognized by driving speeds above 200 km/h [17,18]. Thus, the theoretical required horizontal radii to obtain high speeds might be determined by considering the physical behavior of trains over a curve. This can be achieved by considering the train’s geometrical behavior in terms of maximum track deficiency as well as its physical behavior in terms of maximum permitted speeds along a horizontal alignment [19,20]. Assuming a speed of 250 km/h, the minimum required track radius to achieve this case might be approximated as [17]: (1)RCmin=11.8Vmax2Umax+Imax(m)where Vmax, Umax, and Imax are maximum permissible track speeds (km/h), track Cant (mm) and Cant deficiency (mm). In addition to the minimum curve radius in horizontal alignment is RCmin(m). Imax is calculated using Eq. (2) [17,21]: (2)Imax=Gagegζmax(mm),where ζmax is the highest residual lateral acceleration in (m/s²), “Gage” is track gage, which is the distance among centers both rails, and g is the acceleration of gravity (m/s). It is preferable to have a design that ensures passenger comfort [16]. Hence, a lateral acceleration value of 0.5 was adopted, as it is recognized to have an inverse relationship with passenger comfort. The selection of these values was made based on the application of the Joly and Pyrgidis model, which addresses the lateral behavior of railway vehicles on curves [22–24]. A typical value of 200 m was chosen for Umax. A curved track radius of 2667 m is obtained by substituting variable values in Eq. (1).

Fig. 4 illustrates the geometrical attributes of the G2T-FSO curved track on x and y axes. R_C denotes the track radius, and ϖC1 is the angle between TXC and RXC [9]. The FSO LOS is established between the BS and the train transceiver at TXC and RXC. The BS positioned is set with a horizontal distance dHC1 and vertical distance dVC1 from the track. φ is irradiance angle, ψ is the incident angle at RXC and γ is the tilting angle.

The parameter δ represents the coverage angle along the propagation distance ZC1 between the beam edge and x-axis. Defining PC1 and QC1 as x and y coordinates, the divergence angle θC1 can be determined. Additionally, β and δ represent the coverage angles for TXC1 and RXC1 at the short and long ends, respectively.

According to Fig. 4, the angle between TXC1 and RXC1, which is denoted as ϖC1 is expressed as: (3)ϖC1=LC1+dHC1RC1,where RC1 is curve radius, φ and ψ represent the irradiance and incident angles, respectively, and γ is tilting angle, defined as: (4)γ=θC12+δ.

The angle between the beam edge and x-axis is δ, which is defined as: (5)δ=tan−1⁡(QC1PC1).

Given Q_C1 y-axis coordinate, P_C1 x-axis coordinate might be defined as: (6)PC1=RC1⋅(sin(ϖC1))

Using Pythagorean theorem (7)RC12−PC12=RC1−dVC1+QC1,which indicates that, (8)QC1=RC12−PC12−RC1+dVC1.

Assuming the curved track divergence angle, (9)θC1=β−δ.

Given that, (10)β=tan−1⁡(dVC1dHC1).

Further, the receiver’s irradiance angle is: (11)δ=tan−1⁡(QC1PC1)

Therefore, the divergence angle for the first curved track θC1 is defined as: (12)θC1=tan−1(dVC1dHC1)−tan−1(QC1PC1).

However, the propagation length is formulated as: (13)ZC1=PC12(LC1)+QC12(LC1)

In an FSO link with a clear LOS, side clearance is needed. Referring to Fig. 3 for clearances beside the curved track, where CC1 is the arc length with RC1 radius. Given that CC1/2 is half arc length, the Sagitta of the curved track arc is SC1. SC1′ shows the minimal obstacle-free LOS as half Sagitta. The Sagitta of a circular arc that depicts rail track curve is used to assess track side clearance for LOS to FSO link: (14)SC1=SC1′+dVC1and since (15)RC1′=RC1−dVC1.

Thus, the length of C_C1 of arch within radius R_C1 is defined as: (16)CC1=2(RC1−dVC1)sin(δ).

As shown in Fig. 3, Sagitta SC1 length is the minimum obstacle-free distance for railway track to provide reliable LOS. (17)SC1′=RC1−RC12−CC124=RC1−RC12−(RC1⋅dVC1)2sin2(δ).

Considering a non-diffraction limited geometrical loss, which is a function of divergence angle θ and link Z range defined as [2]: (18)LG(s)(dB)=20∗log10⁡{dR(m)dT(m)+(Z(km)∗θ(mrad))}2dB,where d_R is the receiver aperture diameter, d_T is the transmitter aperture diameter, Z is the LOS range in km, and L_{G (dB)} represents the specific geometrical loss in disciples.

Referring to Eq. (18), and substituting Eqs. (7) and (8) into (18) geometrical loss at curve track is defined as: (19)LG(C1)(dB)=20log10⁡[dgdT+((PC12(LC1)+QC12(LC1))∗(tan−1⁡(dVC1dHC1)−tan−1⁡(RC12−RC1sin⁡(ϖC1)−RC1+dVC1RC1⋅sin⁡(ϖC1.))))].

The optical received power of a single FSO transmitter is typically defined as [25]: (20)PReceived=P7×dR2(dT+θ⋅Z)2×10−αZ10.

dR, dT, and Z are the receiver and transmitter aperture diameters (m), and LOS range in km, respectively. PT and denote to the transmitter power and atmospheric attenuation factor (dB/Km), respectively. Substitute Eqs. (13) and (14) into (20), the derived received power for curved track G2T-FSO link is defined in (20). It is further defined in terms of geometrical properties as by substituting Eqs. (6) and (8) into (20) as displayed in (21). (21)PReceived(C1)=PT×dR2(dT+(tan−1⁡(dVC1dHC1)−tan−1⁡(RC12−PC12−RC1+dVC1RC1⋅sin⁡(ϖC1))2)(PC12(LC1)+QC12(LC1)))2×10(−α(PC12(LC1)+QC12(LC1.))10).

The received power at the receiver for a single transmitter depends on the lens aperture diameter dT(m), transmitter aperture diameter dR(m), channel attenuation factor α (dB/km), link range Z (km), and divergence angle θ (mrad). The equation is given as [25]: (22)PR=PTotal∗dR(m)2[dT(m)+(θ∗Z(km))]2∗10(−α∗Z(km)10).

The photodetector converts the optical signal to an electrical signal. During detection, the signal faces various noise sources: shot noise σShot2, thermal noise σThermal2, and background noise σBackground2. These are used to calculate the SNR as defined by [26,27]: (23)SNR=(RPReceived)σTotal2,where (24)σTotal=σShot2+σThermal2+σBackground2, (25)σThermal2=⟨iThermal2⟩=4KBTBRL, (26)σShot2=2q(R0PReceived)MX+2B,where KB is Boltzmann’s constant, T is absolute temperature, B receiver bandwidth, and RL is equivalent load resistance [28,29]. Moreover, q is the electron charge, R0 receiver sensitivity, and Bit error rate are then computed using [30,31]: (27)BER=Q(SNR),Q(x)=12π∫x∞e−y22dy.

Furthermore, the factor (Q) is calculated and used to estimate the performance of the receiver and its defined as follows [32]: (28)Q=bon−boffσon,

It accounts for the noise power variance when detecting 1 and 0 bits. Another key parameter is the link margin, which evaluates the link’s ability to compensate for scintillation, scattering, and absorption losses. The link margin is defined by [33]: (29)LM=PTotal+|Sr|+GR−LGeometrical−LTx−LRx−LAtmospheric.

Total transmitted power PTtotal, receiver sensitivity Sr, and receiver gain GR are key factors in this equation. Additionally, the parameters LGeometrical, LTx, LRx and LAtmospheric are considered, representing geometrical, transmitter-receiver, and atmospheric losses. It’s assumed that scintillation losses due to high-speed train travel are neglected. To determine the required number of BSs for continuous coverage along the rail track, the calculation is as follows: (30)N.B/1km=1000mMaximum Acceptable Link Range,where acceptable link range is governed by an error-free FSO link with BER of 10⁻⁹. Eq. (6) is used for 1 km rail track and Eq. (7) for a 10 km rail track. (31)N.B/10km=10,000mMaximum Acceptable Link Range.

FSO link establishes an optical link using the earth’s atmosphere. This limits beam transmission due to photon absorption and scattering. Fog and rain attenuation levels were assessed using meteorological data and empirical models. Rain and fog attenuations were categorized by ITU-R code to characterize FSO channel [34]. Table 1 presents the link power budget for the curved track FSO-G2T communication link used in this simulation, with a wavelength of 1550 nm and a data rate of 2.5 Gbps.

The G2T-FSO communication system, shown in Fig. 5, was modeled in OptiSystem with a 4Tx-1Rx configuration. It uses multiple lasers connected to a single OOK modulator to produce multiple beams that propagate along the link.

Fig. 6 illustrates received power levels for curved track G2T-FSO communication link with a radius of R_C1(250) = 2667 m. It shows the effects of different weather conditions on received power levels along the track span.

Moreover, it is divided into two regions, namely, feasible region and infeasible region. The feasible region shows levels for acceptable link performance that achieved a BER of 10⁻⁹, however, the infeasible region shows power levels that are less than the acceptable levels.

In terms of link G2T-FSO range, under clear weather with attenuation of 0.6 dB/km power levels of −29.71, −29.05 and −29.50 dBm were achieved at ranges of 240, 365, 505, and 618 m for 1Tx, 2Tx, 3Tx, and 4Tx NRZ transmitter respectively. Furthermore, under moderate rain attenuation of 6.9 dB/km levels of −28.68, −29.50, −29.73 and −28.99 dBm at ranges of 201, 322, 388 and 399 m for single, dual, triple and quad transmitters, respectively, Moreover, under moderate fog levels of −29.32, −28.83, −28.56 and −29.10 dBm at ranges of 164, 220,260 and 301 m for the aforementioned cases, respectively.

On the other hand, in terms of weather effects on received power, under moderate rain, an average decline of −5.15 and −3.07 dBm occurred for 4Tx and 3Tx, 2Tx, 1Tx, respectively, which is equivalent to an average drop of 22%, 10.36%, 13% and 7.59% for each case, respectively. Furthermore, moderate fog causes an average power level decrease of −13.74, −13.74, −13.79 and −13.81 dBm that amounting to an average of 43.84%, 38.92%, 38.34% and 29.67% for 4Tx, 3Tx, 2Tx and 1Tx, respectively. The Quad Transmitter setup achieved the highest link range and robust performance in various weather conditions. The train communicates through all four transceivers (Tx1, Tx2, Tx3, Tx4) with the BS Transceiver (Rx1), enabling high data rates for passengers.

Acquired SNR levels of multiple receivers under variable weather conditions are illustrated in Fig. 7. It is organized into four subgroups (a), (b), (c) and (d) for single, dual, Triple, and Quad transmitters, respectively. In the case of clear weather conditions, SNR levels of 4.06, 5.39, 4.43 and 4.47 dB at ranges of 240, 265, 505, and 618 m were achieved for the cases above, respectively. Moreover, under moderate rain conditions, levels of 5.66, 4.31, 7.64, and 6.40 dB were achieved at link ranges of 201, 322, 388, and 399 m for 1Tx, 2Tx, 3Tx, and 4Tx, respectively. Lastly, under moderate fog condition SNR levels of 4.73, 6.28, 6.45, and 5.24 dB were achieved at ranges of 164, 220, 260, and 301 m.

As per international Union of Railways (UIC) standards for high-speed railways, the train-to-train communication model achieves optimal performance with a BER = 10⁻⁹ at an SNR of 5 dB or higher, meeting transmission needs for train control and prewarning messages. Simulations show that the 4Tx/1Rx configuration provides the highest SNR as the train nears the BS.

Fig. 8 shows achieved BER levels along with coverage range for a various number of G2T-FSO curved Track transmitters under different weather conditions. The BER is a key parameter for evaluating the performance of data channels, whether over radio, wireless, or wired telecommunications links. It measures the number of errors occurring in the transmitted data, with an acceptable BER typically around 10⁻⁹, indicating very low error rates at the receiving end. The graphs demonstrate acceptable BER levels of 10⁻⁹ were achievable at ranges of 240, 365, 505 and 610 m for 1Tx, 2Tx, 3Tx, and 4Tx, respectively. Additionally, under moderate rain condition, acceptable link performance was achieved at 201, 322, 388, and 399 m. Apart from rain attenuation, moderate fog acceptable link ranges were achieved at distances of 164, 220, 260 and 301 m for 1Tx, 2Tx, 3Tx, and 4Tx, respectively.

An eye diagram, used to assess signal quality, appears as a series of “eyes” between rails. Table 3 shows that the 4Tx configuration performs best under clear, moderate rain, and moderate fog conditions. All transmitter configurations show near-ideal NRZ eye openings at short distances, but eye clarity decreases with greater distance or higher weather attenuation. Moderate fog degrades performance, leading to incoherent eye diagrams at long ranges. The 4Tx configuration exhibits high tolerance to geometrical losses in clear weather but some jitter in moderate rain. The 3Tx configuration also handles weather well but is less effective than 4Tx in moderate fog.

A curved track with a radius of R_C1(250) = 2667 m for link margin analysis in Fig. 9, shows resulted link margins for curved track model were estimated where it illustrates link margins for NRZ for 4Tx, 3Tx, 2Tx and 1Tx under various weather conditions. However, under clear weather link margins of 21, 22.16, 24.32 and 26.59 dB were achieved at ranges of 618, 505, 365, and 240 m for 4Tx, 3Tx, 2Tx and 1Tx utilizing NRZ modulated transmitters. This result shows how track curvature impacts FSO link stability.

For the case of curved track G2T-FSO line, there is only one study that has been conducted by Paudel et al. in [8]. Table 4 shows that 4TX/1RXNRZ G2T-FSO link achieve 124% improvements, respectively.